# What do we mean by 'Equivalent Projective representation"?

by S_klogW
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 P: 19 I know that we say two representations R and R' of a group G is equivalent if there exists a unitary matrix U such that URU^(-1)=R'. But what do we mean by equivalent projective rerpesentations? I've heard of the theorem that the SO(3) group has only 2 inequivalent projective representations. But what does that exactly mean? I am very interested in projective representation because it's projective representation rather than ordinary representation that represents symmetry in Quantum Mechanics since the vector A and exp(id)A represent the same physical state. So does anyone know if there are some books that can serve as an introduction to projective representations?
 PF Patron Sci Advisor Emeritus P: 8,837 Since you haven't received any replies, I will mention that "Geometry of quantum theory" by Varadarajan covers projective representations and their relevance to quantum mechanics. I hesitate to recommend it because I find it very hard to read, but I don't know a better option.
 PF Patron Sci Advisor Thanks Emeritus P: 15,671 I've never done anything about projective representations before, so this post is just a guess. But it would make sense to define first $$Z=\{cI_n~\vert~c\in \mathbb{R}\}$$ Then we define a projective representation as a group homomorphism $$\rho: G\rightarrow GL_n(\mathbb{R})/Z$$ This last group is often called $PGL_n(\mathbb{R})$, or the projective general linear group. Given, $\rho,\rho^\prime$ projective representations, it would make sense to define them equivalent if there exist $U\in O_n(\mathbb{R})/Z$ such that $$\rho(g)=U\cdot \rho^\prime(g)\cdot U^{-1}$$ for all $g\in G$. The complex case is similar.

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